Russell on Real Variables and Vague Denotation

Abstract

As is well known, in the Principles of Mathematics Russell distinguishes between ‘any’ and ‘all’. He uses ‘any’ to understand mathematicians’ inferences using free variables. For example, when one introduces a right-angled triangle xyz in order to prove the Pythagorean theorem, one is saying that each proposition shown is true of any right-angled triangle. In this way, after one concludes that the square of hypotenuse xy is equal to the sum of the square of yz and xz, one can generalize that this is true of all right-angled triangles. The free variables ‘x’, ‘y’, and ‘z’ are used to pick out ‘any triangle’ or an ‘arbitrary triangle’. Russell allows assertions of open formulas in his mathematical logic at least until (and including) the first edition of Principia Mathematica. His discussion of free variables in the first edition of Principia is strikingly similar, in a particular aspect, to the one in Principles.